*Article* **Smart Core and Surface Temperature Estimation Techniques for Health-Conscious Lithium-Ion Battery Management Systems: A Model-to-Model Comparison**

**Sumukh Surya 1,\*, Akash Samanta 2, Vinicius Marcis <sup>2</sup> and Sheldon Williamson <sup>2</sup>**


**Abstract:** Estimation of core temperature is one of the crucial functionalities of the lithium-ion Battery Management System (BMS) towards providing effective thermal management, fault detection and operational safety. It is impractical to measure the core temperature of each cell using physical sensors, while at the same time implementing a complex core temperature estimation strategy in onboard low-cost BMS is also challenging due to high computational cost and the cost of implementation. Typically, a temperature estimation scheme consists of a heat generation model and a heat transfer model. Several researchers have already proposed ranges of thermal models with different levels of accuracy and complexity. Broadly, there are first-order and second-order heat resistor–capacitor-based thermal models of lithium-ion batteries (LIBs) for core and surface temperature estimation. This paper deals with a detailed comparative study between these two models using extensive laboratory test data and simulation study. The aim was to determine whether it is worth investing towards developing a second-order thermal model instead of a first-order model with respect to prediction accuracy considering the modeling complexity and experiments required. Both the thermal models along with the parameter estimation scheme were modeled and simulated in a MATLAB/Simulink environment. Models were validated using laboratory test data of a cylindrical 18,650 LIB cell. Further, a Kalman filter with appropriate process and measurement noise levels was used to estimate the core temperature in terms of measured surface and ambient temperatures. Results from the first-order model and second-order models were analyzed for comparison purposes.

**Keywords:** electric vehicles; stationary battery energy storage system; battery automated system; online state estimation; thermal modeling; first-order model; second-order model; Kalman filtering

#### **1. Introduction**

Lithium-ion batteries (LIBs) have been extensively commercialized as a primary energy storage technology for electric vehicles (EVs), stationary energy storage in the smart grid system and several other consumer electronics. The primary dominating factors of LIBs over other energy storage technologies include high energy density, long lifespan, and declining cost [1–4]. However, from literature and practice, it is noticed that the performance of LIBs as well as the durability and reliability are significantly influenced by the operating temperature. Moreover, excessively high temperatures may cause thermal runaway, leading to fire, smoke and other serious safety hazards to the operators [5–7]. Therefore, the requirement of a battery management system (BMS) has become indispensable for effective thermal management and safety of LIB system, which essentially requires accurate information on the core and surface temperature of each cell [8,9] besides other important states such as state of charge (SOC) [10,11] and state of health (SOH). A few other popular functions of an advanced BMS include cell balancing [12,13], fault detection/diagnosis [14]

**Citation:** Surya, S.; Samanta, A.; Marcis, V.; Williamson, S. Smart Core and Surface Temperature Estimation Techniques for Health-Conscious Lithium-Ion Battery Management Systems: A Model-to-Model Comparison. *Energies* **2022**, *15*, 623. https://doi.org/10.3390/en15020623

Academic Editor: Haifeng Dai

Received: 14 November 2021 Accepted: 4 January 2022 Published: 17 January 2022

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

and some other safety inspection functionalities. Several recent research studies highlighted that the accuracy of estimating cell SOC [15], SOH [16] and remaining storage capacity [17] depends on the accurate estimation of cell temperature as all these states are the function of temperature. Moreover, the Columbic efficiency of a cell is greatly affected by the cell temperature during the charging and discharging period. It is worthwhile to mention that the temperature distribution inside the cell is not uniform, and the core temperature remains higher than the surface temperature during practical application, especially under high charging and discharging current [18]. Typically, the difference between the core and surface temperature varies in the range of 5–10 ◦C [19,20]; however, under high current loading with rapid load fluctuation, it could be even more. Therefore, accurate information on the core and surface temperature is essential to achieving the effective thermal management of an LIB pack besides fault detection. While most of the existing temperature measurement techniques measure the surface temperature directly using physical sensors [21], the measurement of cells' internal temperature is highly challenging when using a physical sensor. Moreover, any high-capacity LIB pack consists of thousands of single LIB cells; thus, installing physical sensors in each cell is not practically feasible from the viewpoint of incremental cost and manufacturing complexity.

To sum up, accurate information on core temperature undoubtedly serves as the essential basis for the thermal management and safety of LIB apart from SOC and SOH estimation whilst it is difficult to measure the core temperature using physical sensors. Therefore, a precise thermal model is crucial to accurate temperature estimation. Moreover, it should be easy to model and computationally inexpensive in order to be implemented in onboard BMS for online prediction of temperature. Several temperature estimation techniques have been proposed by researchers. Typically, a temperature estimation strategy consists of two models, namely, a heat generation model and a heat transfer model [22]. The heat generation model takes physical measurement signals from a cell, typically voltage current, to estimate the total heat generation during charging and discharging. Then, the heat transfer model takes the estimated total heat quantity as model input to predict the temperature of that cell. Depending on the modeling, it can only estimate the core temperature (single-state) or both the core and surface temperature simultaneously (two-state).

Broadly, heat generation models can be classified into three groups, electrochemical models [23–26], data-driven empirical models [27–29] and equivalent circuit models (ECM) [30–32]. Few other researchers have also grouped the heat generation model from the perspective of heat concentration. According to them, the heat generation model could be a concentrated model (all heat is generated at the core), distributed model (heat generated uniformly over the cell) [33] and heterogeneous model [30,34] (due to temperature and current density gradient inside the cell). On the other hand, the heat transfer model can be grouped into finite element analysis (FEA)-based models [32,35–38], lumped multi-node models [27,39–41] and heat capacitive-resistive models [42]. The lumped multi-node model and heat capacitive-resistive models are typically developed based on the analogy between thermal and electrical phenomena. It can be seen that the electrochemical model can produce a very accurate heat generation value provided all model parameters are carefully tuned. However, the electrochemical models are highly complex and computationally expensive. The accuracy of data-driven empirical models highly depends on the experimentally acquired data. Collecting such high-resolution data is challenging, and with the increase in data volume and the number of feature vectors, computational expenses also increase exponentially. On the other hand, an ECM-based estimation model can be designed suitable for online prediction and real-world application by establishing a balance between the computational cost and prediction accuracy. Therefore, ECM-based battery models are extensively used in practice for estimating heat generation in LIB. Further, as far as the heat transfer model is concerned, the heat resistor–capacitor models are easy to develop and computationally efficient compared to FEA-based methods and lumped-parameter multi-node models. The FEA-based methods are highly accurate; however, they come at

the expense of a high computational cost. Resistor–capacitor-based models can be optimally engineered to make a balance between prediction accuracy and computational cost depending on the application requirement. Therefore, heat resistor–capacitor model-based temperature estimation is the prime focus of this present study.

Researchers have proposed different kinds of heat resistor–capacitor models for the accurate and precise internal and surface temperature estimation of LIB. However, the major concern regarding the practical application of any model is its computational cost, the capability of online prediction and suitability for onboard BMS. Detailed studies on the thermal characteristics of different layers inside an LIB cell, modelling complexity and the experimental data requirement have been carried out and are listed in the references section [43–48]. The heat resistor–capacitor models use the analogy between thermal and electrical phenomena, where heat capacity (thermal capacitance) and heat transfer coefficient (thermal resistance) are represented as electrical capacitor and resistor, respectively [43]. So far, a first-order (one thermal energy storage element) and second-order (two thermal energy storage elements)-based thermal models have been reported in the literature for temperature estimation. Second-order models are typically complex and require extensive experiments alongside the knowledge of domain experts during modeling. On the other hand, first-order models are easy to implement, computationally inexpensive and require far fewer experiments. Recently, extensive research effort has been made on second-order thermal models of LIB. However, a comparative study between the first-order and second-order model has not yet been assessed. Therefore, this research study focused on the comparative study to investigate whether it is worth investing in developing and implementing a second-order thermal model for the core temperature estimation of LIB in terms of accuracy, modeling complexity and the experimental requirement and its practicability in onboard BMS. Extensive experiments were conducted for data collection, and the data was further utilized for modeling, validation and comparison purposes. The strategy was to employ an ECM-based heat generation model for both a first-order and secondorder thermal model to determine the total heat generation inside the cell. A Kalman filter (KF) was used in both the cases to improve prediction performance. Then, the estimation results were compared with the measured data to assess the modeling accuracy. Finally, the predicted results obtained from the first-order and second-order model were compared for the purpose of model-to-model comparison.

The remaining portion of the article is subdivided into five sections for better readability, representation and understanding of the readers. First-order and second-order thermal modeling of LIB and the respective temperature estimation strategy are presented in Section 2. The experimental setup and model parameter identification are discussed in Section 3. Temperature estimation using the fusion of the first-order thermal model with KF and second-order thermal model with KF is described in Section 4. Section 4 also includes the comparative study between the first-order thermal model and second-order thermal model in terms of prediction accuracy and modeling complexity. Major findings and concluding remarks are drawn in the conclusion in Section 5.

#### **2. Thermal Modeling and Temperature Estimation Strategy**

Commercially, LIBs are available in many different form factors such as prismatic cells, pouch cells [49] and cylindrical cells. Among these, cylindrical cells are widely used in large-scale high-power applications. However, the cylindrical cell has worse thermal heat dissipation, and the spiral format leads to a big thermal gradient inside the cell. Therefore, the thermal modeling of a 18,650 cylindrical LIB cell is considered in this study, considering the necessity of effective thermal management of cylindrical LIB. The mathematical analysis and the fusion of KF with these thermal models for core and surface temperature estimation are presented in this section. The aim is to provide a guideline for selecting an appropriate thermal model for online prediction with an optimum computational cost suitable for onboard low-cost BMS. As previously discussed in the introduction section, the temperature estimation model consists of one heat generation model and a heat transfer model, where the heat generation model provides input to the heat transfer model. Therefore, the modeling strategy and mathematical analysis of the ECM-based heat generation model are considered here as well.

#### *2.1. Heat Generation Model*

The Electric Circuit Model (ECM) [50]—based thermal estimation model has been reported to estimate the total heat generation inside the LIB cell by several researchers. So far, electrochemical modeling has demonstrated the best performance in capturing the nonlinearities of LIB, while at the same time, it is the most complex to model. Capturing the high degree of nonlinearities higher-order ECM is required; however, the computational cost and modeling complexity increase with the increase in model order. Yet the major advantage of ECM is that a balance between the modeling complexity and model accuracy can be achieved through optimization with the help of the model order reduction technique [51,52]. Therefore, a 1-RC (first-order) ECM is considered here to quantify the total heat generation. The 1-RC ECM of LIB is shown in Figure 1. The basic strategy used by any ECM-based heat generation model is to mathematically accumulate the heat generation from internal power losses that typically depend on the internal resistance and charging– discharging current level. Again, the heat generation depends on the cell SOC, current level and temperature, as the internal resistances are the functions of these variables.

**Figure 1.** 1-RC ECM (Thevenin's equivalent) model of an LIB cell.

The *VOCV* and *V* in Figure 1 represent the open-circuit voltage and the terminal voltage, respectively. The steady-state DC series resistance, which represents the electrolyte resistance to the lithium-ion transportation, is denoted as *R*<sup>0</sup> in Figure 1. Further, the short transient response is caused by the lithium-ion flow in the solid electrolyte interphase layer, and the anode electrode is represented by polarization resistance (*R*1) and capacitance (*C*1), respectively. These *R*<sup>1</sup> and *C*<sup>1</sup> appear only during the transient period [53]. A 1-RC battery model was considered in this study due to its optimum performance, ease of modelling, low computational cost and adequate accuracy when compared to other higher-order RC models [54,55]. Further, the online determination of heat generation inside LIB with these higher-order models is challenging due to computational cost. For this, Bernardi et al. [56] developed a simplified equation for LIB heat generation calculationthatis suitable for online prediction over other computationally expensive methods such as constant heat generation rate [57], curve fitting technique [58] and Joule's Law [59]-based methods. The governing equation for the total heat generated inside the battery (*Q*) as developed by Bernardi et al. [56] is shown in Equation (1).

$$Q = I(V - V\_{OCV}) \tag{1}$$

The parameters of this equation are also the function of charging–discharging current (*I*), SOC and temperature, which are estimated using the ECM of the cell. Finally, the value of the *Q*, obtained from Equation (1), is used as one of the inputs to the first-order and second-order thermal model for temperature estimation, which is discussed in the following section.

#### *2.2. First-Order Thermal Modelling*

#### 2.2.1. Mathematical Analysis of First-Order Thermal Model

Now, for the first-order model, as is noted by several other researchers, the surface temperature is considered constant throughout the surface of the cell. Heat transport is only along the radial direction, meaning the lateral surface temperature is considered the same as axial direction (cell temperature at two terminals), as reported in [43]. Further, regarding heat transfer, only heat conduction from the core to the surface is considered. Heat exchange between surface and ambient by convection is not considered. The first-order thermal model is depicted in Figure 2.

**Figure 2.** First-order heat resistor–capacitor-based thermal model of LIB.

In Figure 2, the thermal parameters, that is, the heat capacity of the core, heat transfer resistance inside the cell, heat transfer resistance outside the cell and total quantity of heat liberated concentrated from the core, are represented by *Cc* (J/K), *Rc* (K/W), *Ru* (K/W) and *Q* (J), respectively. The unit of each respective quantity is mentioned in the parentheses. The temperature of the core, surface and ambient is represented by *Tc*, *Ts* and *Tamb*, respectively, measured in K. The core temperature at node *Tc* and surface temperature at node *Ts* can be monitored using this model; thus this type of model is also referred to as a two-node or two-state thermal model [22,60].The heat resistor–capacitor model uses the analogy between the thermal and electrical systems, as discussed in the introduction section. Thus, for mathematical analysis, the heat transfer rate is represented by electrical current (*i*), and the branch currents are represented by *ia, ib* in the respective branch, as shown in Figure 2. Therefore, the governing equation of the model can be derived by applying Kirchhoff's Current Law (KCL) at the *Tc* node. The current balance equation at node *Tc* reads:

$$
\dot{a} = \dot{a}\_a + \dot{a}\_b = \dot{Q} \tag{2}
$$

Now, by rewriting Equation (2) in terms of thermal parameters, Equation (3) can be found:

$$Q = C\_c \frac{dT\_c}{dt} + \frac{T\_s - T\_c}{R\_u} + \frac{T\_{amb} - T\_s}{R\_c} \tag{3}$$

By re-arranging Equation (3) we find:

$$C\_c \frac{dT\_c}{dt} = Q + \frac{T\_s - T\_c}{R\_u} + \frac{T\_{amb} - T\_s}{R\_c} \tag{4}$$

Finally, the value of *Tc* can be calculated by integrating Equation (4) with respect to the total heat transfer time while the values of *Ts* and *Tamb* are known. While *Tamb* can be easily measured by employing only one temperature sensor, the measurement of *Ts* with physical sensors in a high-power LIB pack is challenging. Therefore, the alternative solution is to estimate the surface temperature using a temperature estimation scheme. One such estimation scheme is also proposed in reference [8], which estimates *Ts* from known *Tc*.

#### 2.2.2. KF for First-Order Thermal Model

KF is used to estimate and predict an unknown parameter from known parameters. The state model for a KF and the first-order model, as developed in the reference [43,61] and in [62] respectively, are also considered for this study. Now, assuming the state as *Tc,t*, output as *Ts,t* and inputs as *Q* and *Tamb*, The state-space matrices are derived by linearizing Equation (4) in the discrete domain. A linearized version of Equation (4) is shown in Equation (5).

$$T\_{\mathbf{c},t} - T\_{\mathbf{c},t-1} = \frac{Q\_{t-1}}{\mathbf{C}\mathbf{c}} + \frac{T\_{\mathbf{s},t-1} - T\_{\mathbf{c},t-1}}{\mathbf{C}\_{\mathbf{c}}R\_{\mathbf{c}}} + \frac{T\_{amb,t} - T\_{\mathbf{s},t-1}}{\mathbf{C}\_{\mathbf{c}}R\_{\mathbf{u}}} \tag{5}$$

As shown in reference [8], small changes in *Ts* can be ignored. Hence, the term *Ts*,*t*−<sup>1</sup> can be considered as zero.

$$T\_{c,t} = \frac{Q\_{t-1}}{Cc} + T\_{c,t-1}(1 - \frac{1}{C\_c R\_c}) + \frac{T\_{amb,t-1}}{R\_u C\_c} \tag{6}$$

The transfer matrices of the KF-based temperature estimation model can be found by reducing Equation (6) in the form of state models as shown in Equations (7)–(9). Hence,

$$\mathbf{A} = \left[1 - \frac{1}{\mathcal{C}\_{\mathbf{c}} R\_{\mathbf{c}}}\right] \tag{7}$$

$$\mathbf{B} = \begin{bmatrix} \frac{1}{\mathbf{C}\_{\mathcal{L}}} & \frac{1}{\mathbf{C}\_{\mathcal{L}} R\_{\mathcal{U}}} \end{bmatrix} \tag{8}$$

$$C = D = 0\tag{9}$$

#### *2.3. Second-Order Thermal Modelling*

#### 2.3.1. Mathematical Analysis of Second-Order Thermal Model

The condition of non-uniform *Ts* and heat transport in the radial direction through conduction from the core to surface is also considered during the second-order thermal modeling. Additionally, the heat exchange between the surface and ambient is considered in the second-order model, which was not included in the first-order model. Only convective heat exchange between the cell surface and ambient is considered here. Therefore, in addition to the thermal properties of the first-order model, the thermal capacitance of cell case (*Cs*) is also considered. The resulting equivalent circuit of the second-order thermal model using heat resistor–capacitor is shown in Figure 3, similarly to the findings of other studies [8,60,63,64].

**Figure 3.** Second-order equivalent circuit thermal model of LIB.

*Q* needs to be estimated for the same ECM-based strategy mentioned in Section 2.1. To derive the mathematical analysis of the second-order thermal model, heat balance analysis in the core and surface is performed. The heat balance equation at the core and surface is represented in Equations (10) and Equation (11), respectively [8].

$$C\_c \frac{dT\_c}{dt} = Q + \frac{T\_s - T\_c}{R\_c} \tag{10}$$

$$C\_s \frac{dT\_s}{dt} = \frac{\left(T\_{amb} - T\_s\right)}{R\_u} - \frac{\left(T\_s - T\_c\right)}{R\_c} \tag{11}$$

#### 2.3.2. KF for Second-Order Thermal Model

*Tc* could be estimated by re-arranging the coupled ordinary differential equations of the second-order thermal model. Since the thermal model has two thermal energy storage parameters (*Cc* and *Cs*), two governing equations are used to estimate *Tc* in terms of measured *Ts* and *Tamb*.

$$\mathbf{A} = \left[1 - \frac{1}{\mathbb{C}\_{\mathbf{c}} \left(R\_{\mathbf{c}} + R\_{\mathbf{u}}\right)}\right] \tag{12}$$

$$\mathbf{B} = \begin{bmatrix} \frac{1}{\mathcal{C}\_{\mathcal{C}}} & \frac{1}{\mathcal{C}\_{\mathcal{C}}(\mathcal{R}\_{\mathcal{C}} + \mathcal{R}\_{u})} \end{bmatrix} \tag{13}$$

$$\mathbf{C} = \begin{bmatrix} \frac{\mathcal{R}\_{\mu}}{\mathcal{R}\_{\mathcal{C}} + \mathcal{R}\_{\mu}} \end{bmatrix} \tag{14}$$

$$\mathbf{D} = \begin{bmatrix} 0 & \frac{R\_{\varepsilon}}{R\_{\varepsilon} + R\_{\mu}} \end{bmatrix} \tag{15}$$

It is worth noting that *Cc*, *Rc* and *Ru* in the second-order thermal model are the same as *Cp*, *Rin* and *Rout*, respectively, in the first-order model.

#### 2.3.3. Fundamentals of KF

It is worth providing a basic explanation of KF as it is the heart of the temperature estimation scheme discussed here. A KF is a linear quadratic estimator and is mainly used in statistics and control engineering. It outputs the estimates of an unknown state and uses the noise and the inaccuracies of the measured output. Some of the common examples of KF usage include guidance, navigation and core temperature estimation in EVs; the general form of KF is shown below:

$$X\_k = A\_{k-1} X\_{k-1} + B\_{k-1} \mathcal{U}\_{k-1} + \mathcal{W}\_{k-1} \tag{16}$$

$$Y\_k = \mathbb{C}\_k X\_k + D\_k \mathbb{L} I\_k + V\_k \tag{17}$$

where *Xk* is the state of the system (*Tc,t*), *Yk* is the output of the system (*Ts,t*), *Uk* is the input to the system ([*Tamb,t Q*] *<sup>T</sup>*), t presents the state of the system and *<sup>t</sup>*−1 represents the previous state of the system. The block diagram of a KF is shown in Figure 4. It is a robust and simple technique used to estimate data based on its input signal. It uses mathematical modeling of the system and by giving the same input as an actual system, it predicts the output. The measured output from the actual system and predicted output from the mathematical model are then compared to obtain the error. This error is multiplied with Kalman gain and is added to the predicted state to obtain an accurate estimated state [65].

**Figure 4.** Fundamental building blocks of KF based estimation scheme.

#### **3. Experimental Analysis for Thermal Model Parameterization**

An automated battery testing system is the best option to collect battery test data, especially for an LIB, as LIB cells are highly sensitive to voltage, current, temperature and other environmental uncertainties. Therefore, an in-house "Battery Automated System (BAS)" was previously developed by the research group of Smart Transportation Electrification and Energy Research (STEER). The setup was used to invent the constant temperature constant current (CT-CV) charging technique [66,67] and several other prominent research studies in the BMS domain [2,4,8]. The BAS is an experimental setup with a fully programmable test environment control and data acquisition system. A schematic layout of the BAS is shown in Figure 5. The experimental data were then used for the parameter estimation of ECM and thermal modeling, model validation and model-to-model comparison purposes. Interested readers are invited to refer to these papers [2,4,8,66,67] for more details about the experimental setup. However, a brief overview of the experimental setup and test conditions is also mentioned in this section as a quick reference for the readers. The basic idea was to identify the input parameters of the thermal model, that is, heat capacity and heat transfer coefficients, through a steady-state analysis as well as transient experiments based on the nonlinear least square algorithm. The LIB cell was tested at three different temperatures where the internal battery temperature was raised using standard current pulses that were within the permissible limit specified on the manufacturer datasheet to ensure no capacity fade occurred during testing.

**Figure 5.** Schematic layout of the Battery Automated System (BAS).

#### *Experimental Setup*

Battery testing was performed on a 18,650 NMC (Lithium Nickel Manganese Cobalt Oxide) LIB, manufactured by LG Chem. Detailed specifications of the cell as provided by the manufacturer are mentioned in Table 1. A programmable power supply (Model: E36313A) from Keysight and a programmable electronic load (Model: BK8601) from B&K Precision were used for charging and discharging the battery with a predefined charging– discharging current profile. Further, a programmable temperature chamber was used to maintain the *Tamb* based on a predefined set-point. Finally, to control the BAS a MATLAB script-based program was used. A programmable data acquisition system (DAQ) (model DPM66204) from Chroma was used to collect the cell voltage, current and temperature data. Different current profile sat three different ambient temperatures (*Tamb* = 273 K (0 ◦C), 293 K (20 ◦C) and 323 K (50 ◦C)) were used for charging and discharging experiments. Finally, a nonlinear least square algorithm was used for online parameter estimation for developing the ECM and thermal model as demonstrated by Surya et al. [8]. All the model components were designed in MATLAB using three-dimension interpolated look up tables where the feature vectors were SOC, *Ibat* and *Tamb*. The heat generation model and the first-order and second-order thermal model were also developed in the MATLAB/Simulink and Simscape environment. Finally, an extensive simulation study was conducted to collect the simulated core and surface temperature data for further analysis. Simulation results were used for model validation as well as model-to-model comparison between the first-order and second-order thermal models. The core temperature (*Tc*) was estimated using a KF for various patterns of currents that were within the permissible limit specified on the manufacturer datasheet to ensure no capacity fade occurred during testing.


**Table 1.** Specifications of 18,650 LIB cell under test.

#### **4. Results and Discussion**

This research study intended to answer whether it is worth developing a secondorder model instead of a first-order model for online temperature prediction by low-cost onboard BMS, firstly, by developing a first-order and second-order thermal model utilizing battery test data and MATLAB-based online parameter estimation; secondly, by simulating the temperature profile of the cell using the first-order and second-order thermal models subjected to different current profiles. The intention was to investigate the impact of charging–discharging current on the core and surface temperature of the cell. Thirdly, we compared the estimation results obtained from the first-order and second-order models. All simulations were carried out in the MATLAB Simulink environment, where a fixed solver and an appropriate step time were used [62]. Initially, simulation was carried out without employing a KF to deduce the baseline analysis. Figure 6 shows the current profile used for the base case analysis, and Figure 7 shows the plots of estimated *Tc*, *Ts* and the measure *Tamb*. Previously, we measured *Ts* from experiments. By comparing the measured and estimated *Ts* it was observed that both *Ts* were within the acceptable limit, and *Tc* and *Ts* closely followed the current profile, and *Tc* > *Ts* > *Tamb*, as per the expectation, confirming the modeling accuracy.

**Figure 6.** The pattern of the discharging current applied to the cell.

**Figure 7.** The plot of *Tc*, *Ts* and *Tamb* without using KF.

In the subsequent sections, firstly, *Tc* was estimated using the combined first-order thermal model and KF for three different current profiles and ambient temperatures, which are illustrated in Case 1, Case 2 and Case 3, respectively. Secondly, a similar study was also conducted for the second-order model and finally, the results were compared. All experiments were carried out with different current profiles as per the manufacturer's recommendation to ensure no battery health degradation [58]. In all cases, the initial currents were kept high for rapid charging of the cell.

#### *4.1. Case 1: Tamb = 293 K (20* ◦*C)*

At first, *Tc* was initialized to *Ts* in the simulation as initially, the cell was in a thermal equilibrium state. *Tamb* was considered as 293 K (20 ◦C), and a very low value of discharging current was applied for the core and surface temperature to rise. The pattern used in Case 1 is shown in Figure 8, and the plot of estimated *Tc* and measured *Ts* are shown in Figure 9 whereas the difference between the estimated *Tc* and measured *Ts* is shown in Figure 10. It was observed that *Tc* and *Ts* closely followed the current pattern, and the maximum difference between estimated *Tc* and measured *Ts* was noted as 6.8 K, whilst it was also noticed that for the entire duration, *Tc* > *Ts*, and the maximum difference occurred when the current was at its peak.

**Figure 8.** The pattern of the discharging current applied to the cell.

**Figure 9.** The plot of the estimated *Tc* and measured *Ts*.

**Figure 10.** Variation of the difference between the estimated *Tc* and measured *Ts*.

#### *4.2. Case 2: Tamb = 323 K (50* ◦*C)*

In the second phase of the experiments, the temperature of the thermal chamber (*Tamb*) was set to 323 K (50 ◦C). The pattern used in Case 2 is shown in Figure 11. Similar to Case 1, *Tc* was initialized to *Ts* during the simulation here as well. The estimated *Tc* and measured *Ts* are shown in Figure 12. It was observed that the temperatures closely followed the current pattern here also. The maximum difference between *Tc* and measured *Ts* was noted as 7K. The plot of the difference between the estimated *Tc* and measured *Ts* is shown in Figure 13. Similar observations to those made for Case 1 were also noticed here in Case 2 regarding *Tc* and *Ts*.

**Figure 11.** The pattern of the discharging current applied to the cell.

**Figure 12.** The plot of the estimated *Tc* and measured *Ts*.

**Figure 13.** Variation of the difference between the estimated *Tc* and measured *Ts*.

*4.3. Case 3: Tamb =273 K (0* ◦*C)*

During Case 3, the temperature of the thermal chamber (*Tamb*) was set to 273 K (0 ◦C), and *Tc* was set equal to *Ts*. The pattern of discharging current applied to the battery is shown in Figure 14. Figure 15 shows the estimated *Tc* and measured *Ts*. Figure 16 shows the

difference between the estimated *Tc* and measured *Ts*. It can be noticed from Figure 15 that at the beginning the magnitude of *Tc* and *Ts* were very large. This was due to the high value of discharging current during this period. It was also observed that the temperature rise is a slow process due to the presence of thermal resistances (*Ru* and *Rc*). The temperature difference increased as the value of discharge current increased. Therefore, it can be inferred from these observations that the temperature rise closely follows the current through the battery, and the rate of rising of *Tc* was the same as *Ts* for a low value of current. However, for higher values of the current the rise in *Tc* was much higher than that in *Ts*. From these observations, the importance of accurate core and surface temperature estimation alongside the requirement of effective and efficient thermal management to maintain *Tc* under the safe operating limit is evidenced.

**Figure 14.** The pattern of the discharging current applied to the cell.

**Figure 15.** The plot of the estimated *Tc* and measured *Ts*.

**Figure 16.** Variation of the difference between the estimated *Tc* and measured *Ts*.

#### *4.4. Comparison between First-Order and Second-Order Thermal Models*

This section deals with the comparative analysis between the first-order and secondorder thermal based on the estimation accuracy, parameter identification, experimental test requirement and suitability for onboard low-cost BMS. To compare the models, the same values of thermal parameters, current, *Ts*, *Tamb* and *Q* were injected into the thermal models. Similar current profiles to those used in Case 1, Case 2 and Case 3 of the first-order model were also applied to the second-order thermal model. Heat generation was calculated using the same 1-RC ECM as used in the first-order model. Finally, the estimated *Tc* profiles obtained from the first-order and second-order thermal models were compared to analyze the prediction accuracy of these models. Figure 17 shows the current profile used for the comparative study, whereas Figures 18 and 19 depict the difference in *Tc* and *Ts* obtained from the first-order and second-order thermal model, respectively.

**Figure 17.** The pattern of the discharging current applied to both the models.

**Figure 18.** Difference between *Tc* and *Ts* obtained from the second-order thermal model.

**Figure 19.** Difference between *Tc* and *Ts* obtained from the first-order thermal model.

It was observed that the difference in temperatures was larger in the first-order thermal model due to the change in *Tc* and not *Ts*. This is because of the decoupling between *Ts* and Tc, as seen in Equation (3). Moreover, while comparing Equations (7)–(9) with Equations (12)–(14), it was noticed that the output parameter *Ts* in KF showed no dependence on the state *Tc,t*−<sup>1</sup> which is also a major reason behind the estimation error in case of the first-order model. Further, references [43,61] demonstrated that *Cc* and *Ru* of the second-order thermal model have a significant effect on *Tc*. Since these parameters were not present in the C and D matrices of the first-order model, a large increase in *Tc* was observed. The thermal parameter sensitivity analysis, as conducted in references [8,61], also confirmed the same reason behind the difference in temperature estimation by the first-order thermal model. It was found that the difference between *Ts* and *Tc* is increased if the discharge current increases. Hence, for currents with dynamic changes, *Tc* estimation using the first-order model provides a large difference from the second-order model. Further, *Cc* only contributed to the transient part of *Tc*. However, with small changes in *Rc* and *Ru*, a large variation in *Tc* was also observed. The modeling complexity, experimental requirement and computational expenses in the used second-order model were not considerably high compared to the first-order model considered here. A tradeoff between

the modeling complexity and accuracy requirement suggests the implementation of a second-order model is worthwhile for smart BMS, especially for high-power applications of LIB.

#### *4.5. Comparison between First-Order and Second-Order Thermal Models for Higher C Rates*

As discussed in the introduction, the performance of different types of battery models is highly influenced by the value of charging–discharging current. As was already witnessed from the above discussion, the second-order model is more accurate compared to the first-order model. However, it is equally important to assess the performance of the second-order model in a high value of discharge current for almost all practical purposes a high value of discharge current is used. Therefore, a discharge current of 5A was applied to both the first and second-order thermal models to observe the change in *Tc*. and *Ts*. The difference between the estimated *Tc* and estimated *Ts* for the first and second-order thermal models is shown in Figure 20.

**Figure 20.** Comparison between *Tc*-*Ts* for higher C discharge.

It was observed that the error (*Tc*-*Ts*) was higher in the first-order model than in the second-order model. Therefore, it could be concluded that the second-order model can also predict a highly accurate temperature state in practical applications as well.

#### **5. Conclusions**

This paper deals with the core temperature (*Tc*) estimation of lithium-ion 18,650 cell using a Kalman filter (KF). This estimation provides effective thermal management, state estimations, operational safety and the longer useful life of LIB. Initially, a detailed discussion regarding the importance of core and surface temperature estimation was presented followed by a review of the state-of-the-art temperature estimation strategies and thermal modeling of LIB. Equivalent Circuit Models (ECM) of LIB-based heat generation model and heat resistor–capacitor-based thermal models were developed in a MATLAB/Simulink environment. Regarding heat resistor–capacitor-based thermal modeling, one first-order and one second-order thermal model were developed and validated using laboratory experimental data. Further, extensive simulation studies were conducted to demonstrate the influence of battery current and ambient temperature on the core and surface temperature of the LIB cell. The heat transfer equations for a first-order and second-order thermal model were derived, modeled and simulated. KF with appropriate process and measurement noise levels was also used to estimate *Tc* in terms of measured surface (*Ts*) and ambient temperature (*Tamb*). Finally, these results were compared to assess the prediction accuracy

of these models. The difference between the core and surface temperatures was noted as approximately 7 K to 8 Kin the first-order model, whereas it was only about 1 K to 2 Kin the second-order thermal model. *Ts* showed no dependence on *Tc* in the first-order thermal model. Further, the output parameter *Ts* in KF showed no dependence on the state *Tc,t*−1, which is also a major reason behind the estimation error in the case of the first-order model. The thermal capacitance of core (*Cc*) and resistances (*Ru*) of the second-order thermal model have a significant effect on *Tc*. Since these parameters are not present in the C and D matrices of the first-order model, a large increase in *Tc* was observed in the first-order thermal model. Hence, the inaccuracy was only due to the error in *Tc* estimation. The findings are also supported by several other research studies in the domain. Further, the consideration of the thermal capacitance of cell casing and the impact of ambient conditions on the secondorder model were the reasons for high accuracy. Further, the performances of first and second-order thermals were also judged with a high value of discharge current for assessing their performance during practical operation. It was observed that the second-order model performance was highly satisfactory compared to the first-order model even in practical applications typically requiring a high value of discharge current. However, estimating the additional parameters of the second-order model requires more experimental data and time. Moreover, due to the complex mathematical form of the second-order model, it takes more computation time. However, looking at the prediction accuracy and the increasing stringent requirement of highly accurate states of battery, it could be stated that it is worth investing more time, cost and expertise in developing a second-order thermal model for more accurate temperature estimation in LIB. This is especially true for the advanced BMS required for high-power LIB packs used in EVs and grid-tied energy storage alongside highly sophisticated consumer electronics. The discussed second-order thermal of a single cell can be extended to an LIB pack by integrating the thermal gradient and the impact of peripheral cells alongside optimal placing of temperature sensors inside the battery casing to adjust the ambient temperature parameter value in the model. All these aspects will be considered in our future research.

**Author Contributions:** Conceptualization, S.S. and S.W.; methodology, S.S. and A.S.; software, S.S. and A.S.; validation, S.S.; formal analysis, S.S. and A.S.; investigation, S.S. and A.S.; resources, V.M. and S.W.; data curation, V.M.; writing—original draft preparation, S.S. and A.S.; writing—review and editing, A.S. and S.W.; visualization, S.S. and A.S.; supervision, S.W.; All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest. No data/information from Robert Bosch Engineering and Business Solutions Private Limited (RBEI) were used for this work.

#### **References**


### *Article* **Controlled Energy Flow in Z-Source Inverters**

**Zbigniew Rymarski, Krzysztof Bernacki \* and Łukasz Dyga**

Department of Electronics, Electrical Engineering and Microelectronics, Faculty of Automatic Control, Electronics and Computer Science, Silesian University of Technology, Akademicka 16, 44-100 Gliwice, Poland; zbigniew.rymarski@polsl.pl (Z.R.); lukasz.dyga@polsl.pl (Ł.D.)

**\*** Correspondence: krzysztof.bernacki@polsl.pl

**Abstract:** This paper proposes a method to reduce the output voltage distortions in voltage source inverters (VSI) working with impedance networks. The three main reasons for the voltage distortions include a discontinuous current in the coils of the impedance network, the double output frequency harmonics in the VSI's voltage output caused by insufficient capacitance in the impedance network, and voltage drops on the bridge switches during the shoot-through time. The first of these distortions can be reduced by increasing the current of the impedance network when the output VSI current is low. This method requires storing energy in the battery connected to the DC link of the VSI during the "non-shoot through" time. Furthermore, this solution can also be used when the Z-source inverter works with a photovoltaic cell to help it attain a maximum power point. The Z-source inverter is essentially a voltage source inverter with the Z-source in the input. In this paper, the theory behind basic impedance networks of Z-source and quasi-Z-source (qZ-source) is investigated where simulations of the presented solutions and experimental verification of the results are also presented.

**Keywords:** impedance network; Z-source; quasi-Z-source; voltage source inverter; voltage distortions

#### **1. Introduction**

The Z-source impedance network was proposed initially by Peng [1]. This type of DC/DC converter was increasing the input DC voltage that is connected to a single-phase or three-phase bridge voltage source inverter (VSI) which switches were used to store energy in the coils of a Z-source. During shoot-through time, energy is stored when both switches in one of the inverter bridge legs are activated. This is only possible only in zero states of the inverter. The modulation index *M* is restricted to the equation *M* = 1 − *dZ* where *dZ* = *TST/Ts*. The parameters *TST*, *Ts*, and *dZ* represent the shoot-through time, switching period of the inverter, and shoot-through time coefficient, respectively.

For a Z-source, it is essential that the shoot-through time, *dZ* is less than 0.5. A voltage source inverter with a Z-source is known as the Z-source inverter (ZSI). An impedance network can function simply as a DC/DC converter with one additional switch in its output realizing shoot-through time but without an inverter. The input current of the Zsource is discontinuous (discontinuous input current—DIC) so Peng showed the changed structure of the impedance network [2,3]. When a diode usually connected in series with the input is replaced, this structure is called a qZ-source. As a result of this modification, the new quasi-Z-source inverter (qZSI) structure is characterized by a continuous input current (CIC) which has improved the use of an impedance network in photovoltaic (PV) systems [4]. Various methods of improving impedance networks structures have been developed [5] and a suitable example is the switched inductor Z-source inverter (SLZSI) [6]. The benefit of using these improved converters is a higher boost factor of the input DC voltage than in the qZSI. Other existing impedance network structures include the embedded SLZSI [7], an inductor-capacitor-capacitor-transformer Z-source (LCCTZSI) [8,9], and a cascaded quasi-Z-source (CqZSI) [10]. The two-winding magnetically coupled impedance source (MCIS) impedance network with a continuous input current [11] has a

**Citation:** Rymarski, Z.; Bernacki, K.; Dyga, Ł. Controlled Energy Flow in Z-Source Inverters. *Energies* **2021**, *14*, 7272. https://doi.org/10.3390/ en14217272

Academic Editors: Sheldon Williamson and Andrei Blinov

Received: 6 October 2021 Accepted: 1 November 2021 Published: 3 November 2021

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

high boost factor. The impedance network circuit based on three coupled inductors with a delta (Δ) connection is presented in [12] and further developed in [13]. The networks found in references [11] and [12] respectively were functional where an additional switch was used without an inverter. A broad review of the impedance network topologies is presented in [14,15], amongst other newly developed solutions based on impedance networks [16–20]. Additionally, several methods of controlling impedance networks have been considered which can be reviewed in [21,22]. However, the symmetric structure of a Z-source with discontinuous input current due to a diode connected in series (Figure 1), and an asymmetric quasi-Z-source (Figure 2) with maximum boost control is sufficient to show the influence of an impedance network on VSI output voltage distortions and proposed ways of reducing these distortions.

(**b**)

**Figure 1.** (**a**) Non-shoot-through state and (**b**) shoot-through state of the Z-source impedance network with the VSI.

Further investigation of these improved network structures has shown that the power efficiency of these systems including the decreased efficiency of the inverter is lower than the efficiency of basic structures. Owing to this decreased efficiency the real boost factor is also much lower than expected [23]. It is worth mentioning that significant differences in recorded levels of radiated disturbances can be expected depending on the type of impedance network structure used [24]. Unfortunately, additional losses in the switches of the VSI during the shoot-through time are observed when switches are absent in the impedance networks. Comparing the performance of a boost converter [23,25], it can be shown that the VSI with an input synchronous boost converter can have a higher efficiency than the same inverter with an impedance network.

(**b**)

**Figure 2.** (**a**) Non-shoot-through state and (**b**) shoot-through state of the qZ-source impedance network with the VSI.

The basic structures of Z-source and qZ-source impedance networks are utilized today in photovoltaic systems [26]. The main disadvantage of these impedance networks lies in the discontinuous current mode (DCM) where the current in the inductors is equal to zero for a time period during *Ts* where there is a low load of the VSI and a low *dZ* coefficient. This is the main reason for the VSI output voltage distortions as shown in Figure 3a,b. By calculating a sufficiently large inductance of the coils [23,27,28] and selecting an appropriate magnetic material [29] for the lowest load while assuming the value of *dZ*, the current in the coils should not decrease to zero. During operation, it cannot be guaranteed that the load current will be nominal. Thus, the additional current taken from the impedance network is a solution of DCM omitting for a low load current.

Another reason for output distortions is the insufficient capacity of Z-source capacitors. Input current from a VSI bridge is like a "rectified" waveform that is filtered by the LC input network and is approximately the first harmonic of the "rectified" current at 100 Hz. This means that 100 Hz distortion is present in the 50 Hz output waveform as shown in Figure 3c. For the insufficient capacity, the output sinusoidal waveform is left-skewed [23,27]. The third type of VSI output distortions are observed after crossing zero output voltage caused by the additional voltage drops on the switched-on transistors during the shoot-through time (see Figure 3a–c), thus causing oscillations after a change of polarization in the PWM voltage. The impedance network influences the dynamic properties of an entire ZSI [23,27,28] which introduces additional resonant frequencies and the additional damping to the Bode plots of the ZSI. The main objective of this paper is to demonstrate how charging the battery from a DC-link after the impedance network during the non-shoot through times can reduce output distortions caused by the DCM of the impedance network. However, charging a battery with too high a current can lead to distortions of the output voltage after the voltage current is zero crossing and oscillations as the result of the higher voltage drops on the switches during the shootthrough time. Experimental results presented will show how charging the battery for a Z-source decreases the output of total harmonic distortions (THD) even in the case when a sophisticated feedback loop, for example, a passivity-based control (PBC), is used.

**Figure 3.** Inverter output voltage distortions, (**a**) Z-source in DCM using a VSI output filter capacitor *CF* = 1 μF, (**b**) Z-source in DCM using a VSI output filter capacitor *CF* = 50 μF, (**c**) 100 Hz harmonic distortions with a Z-source capacitor *CZ* = 100 μF.

Figure 3 presents the different types of output voltage distortions of the ZSI. In Figure 3a,b, the DCM of the Z-source uses a low load current and ZSI output filter capacitors of *CF* = 1 μF and 50 μF respectively. Figure 3c shows the distortions caused by a 100 Hz current harmonic using a high load current and a Z-source capacitor of *CZ* = 100 μF.

Section 2 presents the basic structures of impedance networks and calculations of the minimum ZSI output current *IOUTrmsmin* that ensure their continuous current mode (CCM). In Section 3 the idea of the inverter with the impedance network charging the battery from the DC link (during non-shoot-through time) to keep the impedance network in CCM is presented. The simulations and results of the experimental verification are presented. Section 4 contains the discussion of what kind of previously presented types of ZSI output voltage distortions can be canceled by the controlled charging of the battery. Section 5 presents the final conclusions.

#### **2. Basic Impedance Networks: Z-Source and qZ-Source**

The Z-source and qZ-source impedance networks shown in Figures 1 and 2, respectively, can operate in different states. Two basic states were taken into account during analysis and these include the shoot-through and the non-shoot-through states. The nonshoot-through state is depicted in Figures 1a and 2a, while the shoot-through state [23,27,28] is shown in Figures 1b and 2b.

The Z-source has a symmetrical structure where the values of the inductors are equal i.e., *LZ*<sup>1</sup> = *LZ*2. Similarly, the values of capacitors are the same, i.e., *CZ*<sup>1</sup> = *CZ*2, and the currents in both inductors are the same, i.e., *iLZ*<sup>1</sup> = *iLZ*2. In the qZ-source, the currents in both coils are the same and are identical to the Z-source coils currents (neglecting the influence of the different parasitic resistances) if coils have equal inductances.

The amplitude of the VSI output voltage *VOUTmax* for the ZSI and qZSI is defined in Equation (1) as

$$V\_{\rm OUTmax} = \eta k\_V' M V\_{\rm DC} = \eta \frac{M}{1 - 2d\_Z} V\_{\rm DC} \tag{1}$$

where *η* is the efficiency, *VDC* is the input voltage, *M* is the VSI modulation coefficient, and *kV'* is the DC voltage boost factor of the impedance network without power losses [23,27,28].

It is assumed that the capacitance *CZ* in the Z-source and qZ-source networks are sufficiently high. The average voltage on the capacitors of the Z-source and the *CZ2* capacitor of the qZ-source are identical to the average voltage *VLZav* on the inductors [23,27,28] given in Equation (2) as follows:

$$V\_{\rm LZ1av} = V\_{\rm LZ2av} = V\_{\rm LZ} = \frac{1 - d\_Z}{1 - 2d\_Z} V\_{\rm DC} \tag{2}$$

The input power *PIN* and output power *POUT* of the VSI connected to the impedance networks for a Z-source or qZ-source can be calculated using Equations (3)–(5):

$$P\_{IN} = V\_{DC} I\_{DCav} = V\_{DC} I\_{LZav} \tag{3}$$

$$P\_{\text{OUT}} = V\_{\text{OUTrms}} I\_{\text{OLITrms}} = \eta P\_{\text{IN}} \tag{4}$$

$$P\_{\rm OUT} = \frac{1}{\sqrt{2}} \eta \frac{M}{1 - 2d\_Z} V\_{\rm DC} I\_{\rm OUTrms} = \eta V\_{\rm DC} I\_{\rm LZav} \tag{5}$$

where *ILZav* is a single inductor current averaged over the fundamental period *Tm*.

For the simplest case of the resistive ZSI load, *RLOAD* the output power can be defined Equation (6) as

$$P\_{OUT} = \left(\frac{1}{\sqrt{2}}\eta \frac{M}{1 - 2d\_Z} V\_{DC}\right)^2 \frac{1}{R\_{LOAD}} = \eta V\_{DC} I\_{LZav} \tag{6}$$

And the average inductor current *ILZav* for the root mean square (rms) value of the inverter output current *IOUTrms* is given Equation (7) as

$$I\_{LZav} = \frac{1}{\sqrt{2}} \frac{M}{1 - 2d\_Z} I\_{OUTrms} \tag{7}$$

The *iLZ* inductor current illustrated in Figure 4a comprises three components. These components are the average current *ILZav*, the current *iLZ2fm* which is averaged in the *Ts* switching period, and the triangle component *iLZ*<sup>Δ</sup> of the inductor current. The current *iLZ2fm* has the double fundamental frequency caused by the envelope of the input current of the VSI bridge in the non-shoot-through time while the triangle component inductor current *iLZ*<sup>Δ</sup> is caused by storing energy in the coil during the shoot-through time and recovering energy in the rest of the switching period (in CCM). A plot of the VSI input current is displayed in Figure 4b.

The inductor current *iLZ* is defined in Equation (8) as

$$\dot{\mathbf{r}}\_{\rm LZ}(t) = \mathbf{I}\_{\rm LZ}\mathbf{u}\mathbf{v} + \dot{\mathbf{r}}\_{\rm LZ2fm}(t) + \dot{\mathbf{r}}\_{\rm LZA}(t) \tag{8}$$

Figure 4 shows plots of a Z-source or qZ-source impedance network coil current and an inverter input current including shoot-through current pulses for cases of maximum and close to zero crossing of the inverter output voltage (in CCM).

This most important harmonic component 2 *fm* of the VSI bridge input current flows through the *LZCZ* circuit of the impedance network as shown in Equation (9). It is assumed that all power losses are within the impedance network including the power losses on the VSI switches during the shoot-through time.

$$i\_{\rm LZh2fm}(abs(i\_{\rm LOAD}(t))) = \frac{4}{3\pi}\sqrt{2}I\_{\rm OILrms}\cos(4\pi f\_{\rm m}t) \left|\frac{1}{1-\left(4\pi f\_{\rm m}\right)^{2}L\_{\rm Z}C\_{\rm Z}}\right|\tag{9}$$

**Figure 4.** A Z-source or qZ-source impedance network (**a**) coil current and (**b**) the VSI input current including shoot-through current pulses (that do not supply inverter) in the case of wide (for the maximum of the output inverter voltage) and short (close to zero crossing of the output inverter voltage) inverter PWM pulses in the CCM.

The triangle component *iLZ*<sup>Δ</sup> of the inductor current *iLZ* in the CCM is calculated approximately with the assumption that a sufficiently low capacitor voltage ripple Δ*VCZ* is approximately equal to 0 and *VCZmax* is nearly equal to *VCZav* for the shoot-through time. The triangle component *iLZ*<sup>Δ</sup> can thus be expressed in Equation (10) as

 $\mathrm{i\_{LZ\Delta}}(t) \approx \frac{V\_{\mathrm{CZ\mathrm{m}}}}{\mathrm{LZ}}t$ ,  $\mathrm{i\_{LZ\Delta}}$ max =  $\frac{V\_{\mathrm{CZ\mathrm{m}}}}{\mathrm{LZ}}T\_{\mathrm{st}} = \frac{1}{\mathrm{LZ}}\frac{1-\mathrm{d}\_{Z}}{1-\mathrm{2}\mathrm{d}\_{Z}}V\_{\mathrm{DC}}$  $d\_{Z}T\_{\mathrm{s}}$   $\mathrm{i\_{LZ\Delta}}$ max =  $\sqrt{2}\frac{1}{\mathrm{L}\_{Z}}\frac{1-\mathrm{d}\_{Z}}{\eta M}V\_{\mathrm{OUTrus}}d\_{Z}T\_{\mathrm{s}}$ 

Consequently, the inductor current can be defined Equation (11) as

$$i\_{LZ}(t) = \left[\frac{1}{2}\frac{M}{1-2d\_Z} + \frac{4}{3\pi}\sqrt{2}\cos(4\pi f\_{\rm mf}t)\right]\frac{1}{1-\left(4\pi f\_{\rm mf}\right)^2\mathcal{L}\_{Z}\mathcal{C}\_{Z}}\left|\left|I\_{\rm OUTms} + i\_{LZ\Lambda}(t)\right.\tag{11}$$

The lowest value of the inductor current is calculated Equation (12) as

$$\dot{i}\_{LZ\text{min}}(t) = \left[\frac{1}{2}\frac{M}{1-2d\_Z} - \frac{4\sqrt{2}}{3\pi}\right] \frac{1}{1 - \left(4\pi f\_m\right)^2 L\_Z C\_Z} \left| \left| I\_{\text{OUTrms}} - \frac{1}{2}\dot{i}\_{LZ\text{max}} \right| \right.\tag{12}$$

As shown in Figure 4a, the requirement for CCM is that *iLZmin* must be greater than 0. This phenomenon is expressed in Equation (13) as

$$\left| \left[ \frac{1}{2} \frac{M}{1 - 2d\_Z} - \frac{4\sqrt{2}}{3\pi} \right] \frac{1}{1 - \left( 4\pi f\_m \right)^2 L\_Z C\_Z} \right| \left| I\_{\text{OUTrms}} - \frac{1}{\sqrt{2}} \frac{1}{L\_Z} \frac{1 - d\_Z}{\eta M} V\_{\text{OUTrms}} d\_Z T\_s > 0 \right. \tag{13}$$

From Figure 5a, the absolute value of load impedance expressed in Equation (14) should be lower in value (but always positive) than the value calculated in Equation (14) for CCM for the assigned parameters: *dZ*, *LZ,* and *CZ*, *M* = 1 − *dZ*.

$$|Z\_{LOAD}| < \frac{\eta M L\_Z}{(1 - d\_Z) d\_Z T\_s} (\frac{M}{\sqrt{2}} \frac{1}{1 - 2d\_Z} - \frac{8}{3\pi} \left| \frac{1}{1 - (4\pi f\_m)^2 L\_Z C\_Z} \right|) \tag{14}$$

**Figure 5.** (**a**) Maximum load impedance, and (**b**) minimum output current, that keeps the impedance network in the continuous current mode.

As shown in Figure 5b, the minimum output current for CCM is given Equation (15) as

$$I\_{\text{OUTrms}} > \frac{\frac{1}{ML\_Z} \frac{1 - d\_Z}{1 - 2d\_Z} V\_{\text{DC}} d\_Z T\_s}{\frac{1}{1 - 2d\_Z} - \frac{8\sqrt{2}}{M3\pi} \left| \frac{1}{1 - (4\pi f\_m)^2 L\_Z C\_Z} \right|}\tag{15}$$

The impedance network (Figure 5b) operates in the CCM for the ZSI load current *IOUTrms* higher than the value calculated from Equation (15) for assigned *LZ* = 1 mH and three parameters: *VDC*, *dZ*, and *CZ*. The modulation index *M* has the assigned maximum possible value *M* = 1 − *dZ*.

In Figure 6, the continuous current mode is illustrated where the output voltage of the ZSI is undistorted.

**Figure 6.** CCM waveforms of (**a**) the *ILZ* coil current, ZSI output voltage, and inverter PWM pulses, and (**b**) the undistorted inverter output voltage.

Figure 7 presents the DCM where two cases can be distinguished. From this figure, the distortions of the output voltage are small when the output voltage is below the maximum. When the output voltage is closer to the maximum, the distortions are higher, and the output voltage maximum is lower than expected. For the large VSI output capacitor the VSI output and PWM envelope voltages are shifted when the large VSI output capacitor e.g., *CF* = 50 μF is used. As shown in Figure 7, the short PWM pulses are undistorted in DCM

while the wide pulses are distorted, and the output voltage is lower. The simulation of a DCM operation using the Z-source is presented in Figure 8 for the third PWM modulation schema [30]. The variables used to obtain the measured plots in Figure 8 are given as: *CF* = 1 μF, *dZ* = 0.3, *M* = 0.65, *RLOAD* = 1000 Ω, 3rd modulation schema.

**Figure 7.** Measured DCM waveforms of the *LZ* coil current, ZSI output voltage, and the inverter's PWM wide and short pulses for *CF* = 1 μF and 50 μF inverter capacitors.

**Figure 8.** Simulated DCM waveforms for inverter *CF* = 1 μF, *dZ* = 0.3, *M* = 0.65, *RLOAD* = 1000 Ω, 3rd modulation schema.

#### **3. Controlled Energy Flow—Charging the Battery**

Similar results of measurement shown in Figure 7 and simulations in Figure 8 demonstrate that further simulations of the controlled energy flow i.e., charging the battery is useful. The basic solution is an efficient multi-input-single-output (MISO) [31] feedback that can decrease total harmonic distortions (*THD*) [23,27]. In addition, MISO feedback can decrease two other types of output voltage distortions [27]. However, for systems supplied by varying the DC supply voltage, for example, photovoltaic cells, the controlled energy flow to the batteries, which keeps the CCM, can be used. It is recommended that the battery is charged with a current that is a function of the difference between the calculated value of *IOUTrmsmin* and averaged (10 Hz low pass filter) VSI output current *IOUTrms* as shown in

Figure 9 (if this difference is negative the charging battery current is equal to zero). The actual difference of these currents *IOUTrmsmin* − *IOUTrms* is recalculated (if positive) to match the required increase of the average *ILZav* current expressed in Equation (7). The battery can be charged only during the non-shoot-through state. Energy from the battery is discharged when *VDC* decreases below the assumed value of *VDCmin*, the Z-source is switched off and the shoot-through pulses are blocked.

**Figure 9.** Proposed idea of the inverter with the impedance network charging the battery from the DC link (during non-shoot-through time), and automatic switching to supplying directly from the battery (the positions of switches are presented in the position of discharging the battery when *VDCmin* − *VDC* > 0).

The idea of this system is presented in Figure 9 (for switches placed in the position of discharging the battery). When the battery returns energy, the following happens: the shoot-through pulses are stopped, and the 48 V battery is connected directly to the VSI. This battery voltage should be higher than the amplitude of the output sinusoidal voltage and the modulation index *M* of VSI is increased i.e., *M*<sup>2</sup> is greater than *M*<sup>1</sup> (Figure 9).

Figure 10a presents the simulated waveforms of the *VDC* changed 24/12/24 V (the border value is set to 15 V) with the described automatic action from Figure 9 but without controlled charging the battery when Z-Source operates in the DCM. The following parameters were used in this scenario: *dZ* = 0.3, *M*<sup>1</sup> = 0.65, *M*<sup>2</sup> = 0.75 and *RLOAD* = 1000 Ω. Figure 10b presents that same operation but with controlled charging of the battery for keeping Z-Source in the CCM. The current charging of the battery is calculated as *IBATT* = *f*(*IOUTrmsmin* − *IOUTrms*) using Equation (15), where *f* is a function of Equation (7). The battery charging current *IBATT* calculated from Equations (7) and (15) should be reduced because too high a value of the battery charging current leads to distortions of the VSI output voltage time after the output voltage is zero-crossing (see Figure 11b). These distortions are caused by the high voltage drops on the VSI switches during the shoot-through time. The presented (Figure 10b) reduction of the output voltage THD from 4.6% to 3% without any feedback loop is quite promising.

(**b**)

**Figure 10.** The waveforms of the DC input and AC output voltages of the ZSI switched from a mode of supplying the VSI from Z-source to the mode of supplying VSI from the battery in case of the low input DC voltage, (**a**) without controlled charging battery for Z-source in the DCM for the low load, and (**b**) with controlled charging battery for Z-source in CCM.

The presented simulations were verified in an experimental model using a 12 V battery (without discharging the battery) charged from the DC during *dBTs* pulses where (*dB* = 1 − *dZ*) (Figure 12). The feedback loop was the IPBC2 type presented in [27]. For the DCM mode of the Z-source, the output voltage distortions can be reduced by additional loading the impedance network by means of charging the battery from the DC link in the non-shoot through times.

**Figure 11.** Inverter output voltage (**a**) without charging battery, (**b**) the battery charging current directly equal to *f*(*IOUTrmsmin* − *IOUTrms*), where *f* is a Equation (7), and (**c**) the battery charging with the reduced value of current.

(**a**)

**Figure 12.** *Cont*.

(**b**)

**Figure 12.** (**a**) The inverter experimental set up and (**b**) inverter output voltage distortions comparison for an IPBC controller where *RLOAD* = 2000 Ω, RMS battery charging currents: *IBATT* = 0 (DCM of the Z-source), *IBATT* = 120 mA and *IBATT* = 200 mA (CCM of the Z-source), *dZ* = 0.3, and *dB* = 1 − *dZ*—battery charging pulses coefficient.

The current source from Figure 9 was simply substituted with resistors. Charging the battery allowed for a substantial reduction of output voltage THD from 2.63% to 0.9%. for *IBATT* = 120 mA, but THD increased to 0.97% for *IBATT* = 200 mA. Further research will be on the use of battery charging current not only to reduce the distortions of the output voltage but also looking for a maximum power point (MPP) when the impedance network is supplied from the photovoltaic cell. The battery charging current can be controlled by the coefficient *dB* for the input current of the impedance network would be closer to MPP.

#### **4. Discussion**

The presented results of the simulation and measurements of the experimental ZSI proved that charging the battery from the DC link between impedance network and VSI in the non-shoot-through time can seriously decrease the ZSI output voltage distortions keeping the impedance network in the CCM. The controlled energy flow solution is particularly predicted for the case of wide variations of the input DC voltage and variations of the load current. The output voltage distortions are decreased even when a strong feedback loop of the VSI is present. The controlled charging of the battery can help in the maximum power point tracking when the ZSI is supplied from the photovoltaic cell and this is the perspective of the further studies. In [23], three types of VSI output voltage distortions were distinguished. The controlled charging of the battery can cancel one of them but setting too high a value of this current increases the other reason for distortions. Charging the battery from the DC link of the ZSI during the non-shoot-through time was not presented yet, however, another approach to the controlled power flow for qZSI with charging the battery connected parallel to the *CZ*<sup>2</sup> capacitor (Figure 2) was presented in [32].

#### **5. Conclusions**

In this paper, a technique has been proposed to reduce output voltage distortions in voltage source inverters connected to impedance networks. The proposed method has been validated using simulations and experimentally under different operating conditions. It was discovered that by connecting a rechargeable battery to a DC link placed between

an impedance network and a VSI and employing proper control of the battery charging current during the non-shoot through time, the output voltage distortions in a system with or without feedback can be reduced when a continuous current mode of the impedance network is forced. However, too high a current charging the battery may increase other types of VSI output voltage distortions presented in Figure 11b caused by high voltage drops on the VSI switches during the shoot-through time. Furthermore, the battery charging current can be controlled to increase the impedance network input current to enable the system to reach the maximum power point when the DC source is a photovoltaic cell. The results presented in this paper thus demonstrate that the proposed method is suitable and can be applied in practice to real-time supply systems.

**Author Contributions:** Conceptualization, Z.R.; methodology, Z.R. and K.B.; software, Z.R.; validation, Z.R., K.B. and Ł.D.; formal analysis, Z.R. and K.B.; investigation, Z.R. and K.B.; resources, Z.R. and K.B.; data curation, Z.R. and K.B.; writing—original draft preparation, Z.R.; writing—review and editing, Z.R. and K.B.; visualization, Z.R.; supervision, Z.R.; project administration, Z.R. and K.B.; funding acquisition, Z.R. and K.B. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work was partly supported by the Polish Ministry of Science and Higher Education funding for statutory activities.

**Conflicts of Interest:** The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.

#### **References**

